There is known a control apparatus for a vehicle that includes (i) an engine serving as a drive power source, (ii) a motor/generator serving as the drive power source and (iii) a mechanically-operated transmission mechanism which constitutes a part of a power transmitting path between the drive power source and drive wheels of the vehicle and which is placed in a selected one of a plurality of gear positions with engagement of a selected one or ones of a plurality of coupling devices. A control apparatus for an automatic transmission, which is disclosed in JP-2015-113932A, is an example of this type of control apparatus. This document discloses that, upon so-called “clutch-to-clutch” shifting operation of the automatic transmission (corresponding to the above-described mechanically-operated transmission mechanism), an output parameter value (e.g., an estimated value of a transmitted torque that is transmitted through a clutch of the automatic transmission) is calculated, by using a mathematical model, for reducing a shock and an unsatisfactory acceleration feeling given to a vehicle operator.